Joseph Fourier, Politician & Scientistby David A. Keston, University of Glasgow, United
Kingdom

Et ignem regunt numeri - Plato

"He will have views and prospects to himself
perpetually soliciting his eye, which he can no more help standing still to
look at than he can fly; he will moreover have various Accounts to reconcile:
Anecdotes to pick up: Inscriptions to make out: Stories to weave in:
Traditions to sift: ..."from "Tristram Shandy" by
Laurence Sterne

The life of Baron Jean Baptiste Joseph Fourier (1768-1830), the mathematical
physicist, has to be seen in the context of the French Revolution and its
reverberations. One might say his career followed the peaks and troughs of the
political wave.

He was in turns: a teacher; a secret policeman; a political prisoner; governor
of Egypt; prefect of Isère and Rhône; friend of Napoleon; and
secretary of the Académie des Sciences.

His major work, The Analytic Theory of Heat,
(Théorie
analytique de la chaleur) changed the way scientists think about
functions and successfully stated the equations governing heat transfer in
solids.

His life spanned the eruption and aftermath of the Revolution;
Napoleon's rise
to power, defeat and brief return (the so-called Hundred Days); and the
Restoration of the Bourbon Kings.

Joseph Fourier was born in 1768 in Auxerre in the département of
Yonne; a town steeped in history. He was orphaned before he was ten years
old and grew up with his aunt and uncle in the same town. On the
recommendation of the Bishop of Auxerre he was given a place at the nearby
École Royale Militaire. Under teachers of the Benedictine order he
showed himself to be a fast and diligent student. He studied mathematics
intensly. In the style of the novels of the period, he is reputed to have
collected the stubs of candles so as to study late into the night, thus
ruining his delicate constitution. It is true to say that he was a sickly
child by the standards of the day, suffering from asthma and insomnia.

His teachers saw him as a possible recruit to their order. Yet what he truly
wanted to do was join the army (in either the artillery or the engineers): due
to his lowly background, his father had been a mere tailor, he was
prevented from doing so. Confounded, he chose to enter the Church. He went
then to the abbey at St. Benôit-sur-Loire to ready himself to take his
vows, meanwhile acting as mathematics teacher to his fellow novices.

This was in 1787. In the background we must imagine the first ripples of
revolutionary discontent beginning to cloud French politics.

Fourier never took his vows. He was ready to, but at about the same time an
order from the Constituent Assembly halted further taking of holy orders
throughout France. Whether this was a disappointment to him is open to
question. Nevertheless the necessary order and practical nature of monastic
life were to exert a strong influence on his life's works.

Fourier returned to work as assistant to his old mathematics teacher, Bonard,
at the École Royale Militaire, later the Collège Nationale, in
Auxerre. The congregation of St. Maur to which the teachers belonged was
exempted from the revolutionary clamp-down on the Church as it was a juring
order.

Under the Commune, education was swiftly reorganised to allow entry to a
broader range of students. This was a continuation, in the revolutionary
manner, of the reforms of the deposed Louis XVI. Modern historians see Louis'
attempts as one of the root causes of the Revolution. The combination of
improved education and a continental recession left a large disgruntled
population.

Fourier stayed in Auxerre for the first four years of the Revolution. By this
time the call to revolution could not sensibly be ignored. The high ideals of
those early years swept him along, but later as France felt the paranoia of
the Terror, Fourier was amongst those who tried to resist.

There had been a trade war raging between Britain and France. When the
inevitable war was declared, on the 1st of February 1793, it was popular, at
least in Paris. The call to revolution became a call to arms under Carnot's
levée en masse. Fourier addressed the
Popular
Society in Auxerre on this subject, urging that the men levied should be
true volunteers. Due to his eloquence he was invited to join the society,
being deputised to organise the levée in the region. This meant being
elected to the committee of surveillance, an organisation later to be used as
a secret police force. The municipality trusted him to carry out a number of
other missions of importance. One of these missions got him into trouble.

By the summer of that year, France was in turmoil; there were set-backs in the
war abroad and in the Vendée a royalist counter-revolution was in full
swing (ironically, partly due to resistance to the unpopular levée).
Fourier was collecting horses for both war efforts and returning from one
mission, passed through Orléans. At the time this was a hot-bed of
sans-culottes activity. Through a series of circumstances the leaders of the
sans-culottes of Orléans were disliked by certain members of the
Convention, including the town's administrator (a Montagnard - supporting the
bourgeois and the advancement their money could bring).

In this atmosphere Fourier spoke on the behalf of the leaders of the town's
sans-culottes. The local administrator complained to his Parisian connections
and eventually Fourier was arrested on a charge of being an Hébertist
at a time when Robespierre was increasingly paranoid about other Jacobin
clubs. Our hero was imprisoned only to be released after Thermidor 9 (1794)
when Robespierre was beheaded.

Due in part to the excesses of the Terror, in which many would-be teachers had
lost their heads, there was a political will for a college to train new
teachers. The École Normale was founded for this purpose. Fourier was
nominated as a pupil for the newly formed school. Most striking to the modern
reader are the guidelines on which the school was run: the lectures were to be
delivered standing; with no prepared notes; the lecturers were to accept
questions from the floor when they arose; and the lecturers themselves were
expected to be actively researching in the field they taught! The first
séance was in January of 1795. Fourier accepted with alacrity perhaps
because it offered him a chance to escape Auxerre for the 'big lights' of
Paris. He made good use of his time there in the first half of 1795 and was
certainly one of the pupils best able to cope with the class of lectures
offered. The political pendulum was on a reactionary swing; in September 1795
Fourier was rearrested, again associated with the events in Orléans
and his subsequent behaviour, this time being charged as a Robespierrist!
This second imprisonment seems to have been the more serious. Fourier writes
later that he truly believed he was to die. Friends and
colleagues
pleaded on his behalf. Again it was to be outside events that saved him:
after Napoleon's whiff of grape shot there was another amnesty and
Fourier was released. Using
connections made at the École Normale he went into teaching
mathematics at the École Centrale (later to become the École
Polytechnique)

On the 16th of May 1798, an armada of 180 ships sailed from Toulon. On board
were 30,000 soldiers and sailors; Napoleon Bonaparte, his generals and
officers; and an entourage of 165 scientific and literary intelligentsia, the
so-called Legion of Culture, among them Fourier. Their destination was known
only to Napoleon, and a few of his most trusted friends.

Though Nelson, with the British fleet, was hunting the Mediterranean for him
Napoleon seemed unperturbed, concentrating on the campaign ahead. For the
crews and passengers the story was quite different; the stress was tangible.

It was in this climate that Fourier must first have met Bonaparte. As a
professor at the École Polytechnique, Fourier ranked with Napoleon's
staff and would certainly have been involved in the institutes. These
were the soirées for the officers and savants which Napoleon instigated
and took part in. Fourier almost certainly met General Kléber
(Napoleon's 'right hand' in Egypt).

After the capture of Malta the fleet continued towards Egypt. The mission was
now obvious, to "liberate" the people of Egypt from their brutal
'uncultured state'. The armada landed at Alexandria, on the 1st of July 1798
(and took it three days later). Thereafter the campaign was plagued with
problems. Only a few days after taking Cairo from the Marmelukes, the French
fleet was scuttled off Aboukir Bay. The French were effectively exiled; what's
more, the people they came to free did not seem to favour their presence!

Even so, Napoleon (with his organising spirit) set to work on the situation in
Cairo. He formed the divan - a kind of municipal council - including a
French observer (a post which Fourier would later fill). What was more
important for Fourier, Napoleon created the Institute d'Égypte. The
founding precepts were noble indeed: the nurture of the sciences in Egypt; the
collection of data historical, statistical, etc.; and as a research and
development arm of the army. The latter aim was surely due to Bonaparte
himself.

The campaign in Egypt faltering and dire news arriving from home, Napoleon
returned to France where he was elected Consul. He gave command of Egypt to
Kléber. This was short-lived as Kléber did not endear himself to
the Egyptians and was assassinated. Fourier read the funeral address. The new
commander Menou knew of Fourier's work at the Institute and had him put on the
Divan among other jobs of lesser importance, the sum of which however
made Fourier civilian governor of Egypt.

Harassment by the British Expeditionary Force became a threat in March of 1801
when General Abercrombie landed at Aboukir Bay. The French defence was
outmanoeuvred at Alexandria. In Cairo the members of the Institute decided to
try to return to France. They travelled to Alexandria but as they left port
their ship was stopped by the British blockade under Sir Sidney Smith.

In a gesture characteristic of the era Sir Sidney gave the savants safe
passage to Toulon (but not their research
materials
). Fourier was back in France and it was cold!

Soon after his return Napoleon wrote to him explaining that the first prefect
of Isère, Ricard de Séalt, had recently died, and that Fourier
would be ideal for the job. Fourier must have been displeased; he had been
expecting to return to Paris and the lively scientific discourse of savants of
his own calibre. Disappointed though he was, Fourier was cannier than to
refuse Bonaparte.

Ricard had not left the affairs of the département of Isère in
good order. The administration (with its centre at Grenoble) was a shambles,
so that Fourier had to attend to virtually everything himself. He created
several posts, which ensured that many of his friends were given jobs in the
region. Fourier had many responsibilities: as prefect he was both a public
figure and a governor. The demands of the Minister of the Interior (his
immediate superior) for statistical information were unending. This
information could be anything from the price of sheep at market to the number
of known dissenters and members of religious sects. The darker side of the job
was the suppression of anti-government literature and censorship of the local
newssheet.

He was expected to take care of matters of diplomacy where they concerned
Isère; he organised the visits of the king of Spain and Pope Pius VII
to the region as well as that of Napoleon.

By 1809 the Parisian government were pressing for completion of the research
work on the Egyptian Campaign, which Fourier was to compile. At one stage
Fourier was researching and carrying out his prefectural responsibilities and
readying the definitive version of the Cairo Institute's report for
publication, the "Déscription de l'Egypte" all at the same time.
He got round this by requesting leave in order to complete the Egyptian work.
He certainly used some of this time to continue his experiments.

He was decorated twice: once as a chevalier in Napoleon's Légion
d'Honneur, in 1804, and on completion of the Déscription de l`Egypte,
in 1809, as a baron (a title which commanded a healthy pension).

His major public works were the draining of a large area of marshland at
Bourgoin and the partial construction of a road to Turin. The former may sound
simple enough, but since there were many clashing interests between the
farmers around the land, the peasants living there and the local nobility,
Fourier was forced to visit virtually all these people to bargain with them
for their cooperation.

A direct road to Turin was an obvious step now that Italy had been annexed.
Fourier finally received permission from the Minister of the Interior to build
this road; the minister had been sceptical initially. It was not completed
under Fourier because of the
abdication of Napoleon in April 1814.

This event placed Fourier in a
predicament, for Napoleon's road to exile on Elba ran through Grenoble.
Fourier wrote to beg that the route be changed, giving the excuse that there
was civil unrest in Grenoble and Bonaparte's passing would stir up trouble.
Of course he was more intent on avoiding the embarrassment of seeing his
friend defeated. He need not have worried; Napoleon's entourage did not go
through Grenoble in the end.

Circumstances were different a year later, when Napoleon escaped from Elba.
This time he headed straight for
Lyons on
a route that must pass Grenoble. Fourier was warned and therefore prepared a
token defence of the town at one gate and left by the other, heading towards
Lyons (ostensibly to warn the Bourbons of the danger). He then returned, to be
intercepted by Napoleon's men. Napoleon was displeased at Fourier's
"desertion" but perhaps remembering his own desertion of his friend
in Egypt he gave Fourier the prefecture of Rhône.

When Fourier realised the vengeful nature of the government of the Hundred
Days, he resigned this post. Napoleon was defeated at Waterloo and sent into
exile on St. Helena. The following upheaval in France was mirrored in
Fourier's life; he lost his barony, his pension and his reputation. Out of
favour and out of work, he had to sell many of his personal belongings to
survive. The Bourbons now re-established pettily excluded anyone associated
with Napoleon.

In the end Fourier was offered a job in the Bureau of Statistics of the
département of the Seine (including Paris) by the comte de Chabrol de
Volvic: the prefect of that department and a friend of Fourier's through the
École Polytechnique and Egypt. This paid enough to live on while
giving him time for research.

When the
Académie des Sciences had open elections in 1816, Fourier won the
vote; the King however would not approve his membership. The next opportunity
to join was after the death of a member, in
1817. He
won the election with ease. This time there were no objections. He remained a
member for life.

Fourier had been interested in the phenomenon of heat transfer from as early
as 1802. It would be anecdotal to think that his return to France's cold
shores from Egypt was the cause, though from accounts of his years in
Isère, it is conceivable that he had contracted
myxedema
in Egypt; a disease that would make life in the cold of the Jura
mountains all but unbearable.

There are two distinct problems in any description of heat propagation:
the steady state (where a steady heat is supplied to the body concerned
which eventually reaches an equilibrial distribution); and the time-dependent
cooling of a body in air of a fixed temperature.

His first attempts were in the nature of an abstract model: no surface heat
transfer; discrete bodies; and heat transferred by some shuttle mechanism from
hot to cold. Starting with two bodies of equal mass (and of the same material)
and adding more bodies thereafter, he hoped to generalise still further to
n bodies arranged in a straight line, eventually setting n to
infinity. The resulting model was not satisfactory. With hindsight we can say
that he had not included any terms that described why heat was conducted at
all: he had only an ad hoc conductivity coefficient.

He was stuck until 1804, when Biot visited him in Grenoble. Biot was himself
working on heat propagation in solids and in his work he separated the
treatment of the interior heat transfer and the surface effects. What is
more important, his work dealt with continuous bodies. With these hints
Fourier was well placed to begin research. The simplest problem of this kind
was the thin bar (prism), heated at one end and cooled to a steady temperature
at the other. Here the model was one-dimensional and surface cooling was
always normal to the bar.

Fourier would later complain that Biot did not give references to the works of
Amontons or Lambert that both discuss the temperature curve along a bar heated
at one end. The former assumed a linearly decreasing curve; the latter argued
for a logarithmic
effect.
Fourier's draft paper of that time made it clear that the linear decrease
would imply no net loss of heat to the surrounding air, an obvious flaw.
He defined:

Heat flow per unit area = K (dy/dx)

where K is coefficient of conductivity and (dy/dx) is the
temperature gradient. Then he described three thin slices of the bar at
x, x + dx, and x + 2dx at temperatures y, y'
, and y" respectively. So that:

and this is identified with the net rate of heat loss to the air = 8lhy
dx (where h = coefficient of external conductivity and
the prism has a square cross-section, 2l × 2l). In the steady
state these are equated to give:

y`` = (2h/Kl)y,

where y`` is the second (ordinary) derivative w.r.t. x
.

Next in complexity to the thin bar came the thin ring (annulus). This again
was effectively one-dimensional. Perhaps feeling more adventurous he
considered the cooling problem for the ring too.

In December of 1807 Fourier read a long memoir on "the propagation of heat in
solids" before the Class of the Institut de France. It concentrated on heat
diffusion between discrete masses and certain special cases of continuous
bodies (bar, ring, sphere, cylinder, rectangular prism, and cube).

The diffusion equation used can be stated in three dimensions. The paper was
never published since one of the examiners, Lagrange, denounced his use of the
Fourier Series to express the initial temperature
distribution.

In the case of the thin bar Fourier used the same method as he had in his
earlier draft. In it he still used the "slices" method of deriving the flux.
Later (in his Essay of 1811 and in his book Analytical Theory of Heat)
the slices became mathematical sections, thus resolving difficulties with the
description of the heat flux. The problem had been that by using slices -
albeit infinitesimally thin ones - he had assumed a temperature jump (the heat
in any one slice had to come from its immediate neighbours). This was not
physically realisable (not even in theory), thus became a major flaw. The
introduction of mathematical sections in place of slices avoided this trouble.
Unfortunately detractors (chiefly Biot,
Laplace and
Poisson) did not seem to be aware of the significance of this change from
"temperature difference" to "temperature gradient". This was one of the
criticisms was levelled at Fourier's research after 1807 (the other criticism
often made was the difficulty with periodicity).

The examiners pointed out that his works did not give
Euler and d'Alembert their due. This Fourier conceded, though he still claimed
that his work on trigonometric series was independent since he had had no
access to the relevant mathematical works in Grenoble. Certainly his treatment
of these series was original.

One omission which did cause him trouble related to Biot's 1804 paper. It
seems that Fourier had sent Biot an early copy of his memoir: one with no
reference to Biot's part in the development. Since they were both working in
the same field, there was a certain amount of rivalry. Feeling slighted, Biot
wrote scathingly about Fourier's memoir in the "Mercure de France", a
public journal. Fourier was outraged. They remained enemies throughout their
careers and took great pleasure deprecating the other's achievements.

In 1809, Poisson, a friend of Biot's and another scientist researching heat
propagation, wrote in the Nouveau Bulletin des Sciences (of which he was
editor) about the state of knowledge in the field. He cited Biot's work
generously; while his remarks about Fourier's investigations deliberately
disregarded the very considerable discoveries Fourier had made in the various
special cases described in his memoir. The whole article seemed calculated to
insult.

The last section of the 1807 memoir was a description of the various
experiments which Fourier had undertaken. They follow what seems to be the
chronological progression of his research.

Having treated the problem of the thin bar in 1804, this last section began by
describing the heated annulus: in the steady state and then as it cooled.

His equipment was basic but effective: a polished iron ring of
diameter ~30cm held in place by wooden supports and heated by an
adjustable Argand burner. Six holes were drilled halfway into the ring, four
of which held thermometers on the
Réamur
scale (the space between the ring and the thermometer being filled with
mercury - as were the other two holes). To achieve the steady state one point
in the ring was heated while rest of the ring was allowed to radiate heat
freely and on the whole results agreed very well with Fourier's theory.

For the time-dependent case the ring was placed halfway into a furnace and
then removed to an insulating bath of sand. The initial distribution was of
uniformly hot around one half and cold around the other. The curve was then
seen to iron out as heat flowed from hot to cold. Fourier's theory suggested
that very soon this would resolve even further to give a simple sinusoidal
pattern that would gradually dampen until the ring was at a uniform
temperature. The experiments bore him out: measurements were more troublesome
but conclusive. Those small discrepancies that were there, were probably due
to the unreliable nature of the thermometers being used (thermometers, of the
type Fourier used, tended to be affected by atmospheric pressure both in their
manufacture and in their application).

Fourier treated the initial distribution of temperatures around the ring as a
superposition of many simple sinusoids that varied from peak to trough to peak
an integer number of times along the circumference of the ring. He reasoned
that the higher frequency sinusoids would damp out rapidly. A sinusoid with
twice the frequency would imply that the distance between hot peak and cold
trough was halved and on top of that the temperature gradient would be
doubled. As a result, a sinusoidal distribution with twice the frequency would
dampen at four times the rate.

Parallel to his theoretical development, Fourier then tackled the experimental
rate of cooling of a uniformly heated sphere. He approached this by heating a
small polished iron sphere, then allowing it to cool. The sphere was specially
drilled so that thermometer readings could be taken at its centre. He found
that varying the method of heating had little effect, while blackening the
surface would approximately double the cooling rate. To finish, he compared
the rates of cooling of a cube and a sphere (where the cube had sides the same
length as the diameter of the sphere). This he did by repeating the above
experiments with both solids under the same conditions.

The results of both these were less than impressive: the substantial
experimental errors were put down to deficiencies of the equipment used
especially the thermometers.

In 1810, the Institut de France announced that the Grand Prize in Mathematics
for the following year was to be on "the propagation of heat in solid bodies".
The ideal title for Fourier. The choice was certainly influenced by the
political chicanery of
Laplace and
Monge, supporters of Fourier's cause, and Lagrange, one of his detractors.

The Commission of
judges
was named as Lagrange, Laplace, Malus, Haüy, and Lacroix.

The essay repeated the derivations from his earlier works, while correcting
many of the errata. The only major change was in his treatment of the flux, as
mentioned previously. In 1812, he was awarded the prize and the sizeable
honorarium that came with it. He won the prize, but not the outright acclaim
of his
referees. They accepted that Fourier had the right equations, but felt
that his methods were not without their difficulties.

His prize winning essay would in the normal course of things now be published.
Due to his heavy prefectural duties, he was unable to see this happen. It was
only when he had returned to Paris for good that he could work on getting what
was now The Analytical Theory of Heat printed. Even then the Theory was
only published in 1822 (due to the intransigence of Delambre then the
permanent secretary to the Institute - perhaps at the bidding of Lagrange).
This book contained the analytic side of Fourier's essay only. A companion
volume that would cover his experimental work, problems of terrestial heat,
and practical matters (such as the efficient heating of houses) was never
completed. The actual text of his original essay appeared in two parts in the
Memoirs of the Académie des Sciences in 1824 and 1826. This was only
after Fourier had been elected permanent secretary.

The germ of the idea of Fourier Series is to be found, with hindsight, in
the works of Leonard Euler. Euler was interested in interpolation. In a
problem with origins in planetary perturbations, he approached and solved
problems of the form

f(x) = f(x-1) + F(x)

the solutions were the well-known trigonometric series.

Euler, too, produced the correct Fourier sine expansion, though from
erroneous sources. Lagrange delivered this same result in the context
of sound wave propagation.

In 1754, d'Alembert, working on an astronomical problem, obtained a
trigonometric cosine expansion for the reciprocal of the distance between two
planets in terms of the angle between the vectors from the origin to the
planets: this saw the introduction of the integral expression for the Fourier
coefficients.

There was great debate on the true meaning of the concept of the function.
With many infinite trigonometric series appearing in a variety of situations,
the stage was set for Fourier to break new ground with his consistent use of
the equations named after him.

These same names were associated with another problem of the day - the
vibrating string. As early as 1753, Bernoulli advocated a
trigonometric series solution, on physical grounds. While Euler and d'Alembert
favoured a functional solution, despite problems at t = 0 (i.e., a
corner in a supposedly differentiable equation). Euler could not accept
Bernoulli's solution because he could not reconcile the periodicity of such a
series with the obvious non-periodicity of the physical problem. He was
certain that such trigonometric series could not represent non-periodic
functions at all. His mistake was his uncertainty about the difference between
algebraic generality over the whole real line and geometric generality over
the fixed length of the string. Fourier resolved this problem of periodicity
in Bernoulli's favour in his memoir of 1807.

Fourier's Analytic Theory of Heat derived and justified the basic equations of
heat propagation. Part of its impact was that it did not fit into the scheme
of "rational and celestial mechanics", yet it retained a mathematical
simplicity.

Predominantly mathematical in flavour, the outstanding nature of the work
stemmed from its use of analysis. Partial differential equations were used to
represent physical phenomena and then initial and boundary conditions were
applied as had been done in many special cases by others. Fourier went
further, he distinguished between actions in the "interior" and those at a
"surface" boundary (expressing each as a separate equation). He chose
coordinate systems freely to take advantage of symmetry arguments.

He added explicit statements of initial conditions so that explicit
calculation could be compared with experimental test.

The techniques he used could not have arisen without his contribution: none of
his contemporaries was as secure in their use of separation of variables nor
as confident in infinite series solutions. The beauty of these techniques was
- and remains - their wide applicability.

When Victor Hugo wrote in Les Misèrables:

There
was a celebrated Fourier at the Academy of Science, whom posterity has
forgotten; and in some garret an obscure Fourier, whom the future will recall.

Part 1, Book III, Chapter I

he did not have the benefit of two centuries' hindsight - his Fourier in a
garret is Charles Fourier, a political philosopher. Much of the mathematics of
the nineteenth century (and indeed the twentieth) is heavily indebted to the
Analytic Theory:
definite
integrals defined as sums; the theory of infinite determinants; and uniform
convergence. From a work which was, at least in spirit, physical this was and
still is breathtaking. Joseph Fourier's legacy will certainly not be forgotten.

I acknowledge that in creating this page I have heavily leant on
a number of very readable sources. The Bibliography